Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad Chapter 11 Inductors.

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1 Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad Chapter 11 Inductors

2 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES Become familiar with the basic construction of an inductor, the factors that affect the strength of the magnetic field established by the element. Be able to determine the transient (time-varying) response of an inductive network and plot the resulting voltages and currents. Understand the impact of combining inductors in series or parallel.

3 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INTRODUCTION In many ways, the inductor is the dual of the capacitor; that is, the voltage of one is applicable to the current of the other, and vice versa. Like the capacitor, the inductor exhibits its true characteristics only when a change in voltage or current is made in the network.

4 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAGNETIC FIELD A magnetic field (represented by concentric magnetic flux lines) is present around every wire that carries an electric current. The direction of the magnetic flux lines can be found simply by placing the thumb of the right hand in the direction of conventional current flow and noting the direction of the fingers.

5 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAGNETIC FIELD FIG. 11.10 Determining the direction of flux for an electromagnet: (a) method; (b) notation. FIG. 11.3 Flux distribution for two adjacent, like poles. FIG. 11.13 Defining the flux density B.

6 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAGNETIC FIELD In the SI system of units, magnetic flux is measured in webers (Wb) as derived from the surname of Wilhelm Eduard Weber. The applied symbol is the capital Greek letter phi, Φ. The number of flux lines per unit area, called the flux density, is denoted by the capital letter B and is measured in teslas (T) to honor the efforts of Nikola Tesla.

7 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAGNETIC FIELD The flux density of an electromagnet is directly related to the number of turns of, and current through, the coil. The product of the two, called the magnetomotive force, is measured in ampere-turns (At) as defined by:

8 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAGNETIC FIELD FIG. 11.14 Milligaussmeter.

9 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE In total, therefore, inductors are designed to set up a strong magnetic field linking the unit, whereas capacitors are designed to set up a strong electric field between the plates. FIG. 11.16 Defining the parameters for Eq. (11.6).

10 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Inductor Construction It is also sensitive to the number of turns of wire in the coil as dictated by the following equation and defined in Fig. 11.16 for two of the most popular shapes:

11 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Inductor Construction FIG. 11.18 Air-core coil for Example 11.1.

12 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Types of Inductors FIG. 11.20 Inductor (coil) symbols.

13 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Types of Inductors FIG. 11.21 Relative sizes of different types of inductors: (a) toroid, high- current; (b) phenolic (resin or plastic core); (c) ferrite core.

14 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Types of Inductors FIG. 11.23 Variable inductors with a typical range of values from 1 mH to 100 mH; commonly used in oscillators and various RF circuits such as CB transceivers, televisions, and radios.

15 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Practical Equivalent Inductors FIG. 11.24 Complete equivalent model for an inductor. FIG. 11.25 Practical equivalent model for an inductor.

16 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTANCE Measurement and Testing of Inductors FIG. 11.27 Digital reading inductance meter. (Courtesy of B+K Precision.)

17 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCED VOLTAGE v L FIG. 11.28 Generating an induced voltage by moving a conductor through a magnetic field. Faraday’s law of electromag netic induction.

18 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCED VOLTAGE v L FIG. 11.29 Demonstrating Faraday’s law.

19 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE A great number of similarities exist between the analyses of inductive and capacitive networks. That is, what is true for the voltage of a capacitor is also true for the current of an inductor, and what is true for the current of a capacitor can be matched in many ways by the voltage of an inductor.

20 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.31 Basic R-L transient network.

21 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.32 i L, y L, and y R for the circuit in Fig. 11.31 following the closing of the switch.

22 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.33 Effect of L on the shape of the i L storage waveform.

23 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.34 Circuit in Figure 11.31 the instant the switch is closed.

24 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.35 Circuit in Fig. 11.31 under steady-state conditions. FIG. 11.36 Series R-L circuit for Example 11.3.

25 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.37 i L and v L for the network in Fig. 11.36.

26 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS Since the current through a coil cannot change instantaneously, the current through a coil begins the transient phase at the initial value established by the network (note Fig. 11.38) before the switch was closed. It then passes through the transient phase until it reaches the steady-state (or final) level after about five time constants. The steadystate level of the inductor current can be found by substituting its shortcircuit equivalent (or R l for the practical equivalent) and finding the resulting current through the element.

27 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 11.38 Defining the three phases of a transient waveform.

28 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 11.39 Example 11.4.

29 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 11.40 i L and v L for the network in Fig. 11.39.

30 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.41 Demonstrating the effect of opening a switch in series with an inductor with a steady-state current.

31 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.42 Initiating the storage phase for an inductor by closing the switch.

32 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.43 Network in Fig. 11.42 the instant the switch is opened.

33 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE RELEASE PHASE

34 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.45 The various voltages and the current for the network in Fig. 11.44.

35 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.46 Example 11.6.

36 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.47 Determining R Th for the network in Fig. 11.46.

37 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.48 Determining E Th for the network in Fig. 11.46.

38 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.49 The resulting Thévenin equivalent circuit for the network in Fig. 11.46.

39 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.50 The resulting waveforms for i L and v L for the network in Fig. 11.46.

40 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.51 Example 11.7.

41 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.52 Thévenin equivalent circuit for the network in Fig. 11.51 for t ≥ 0 s.

42 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t = L/R Th FIG. 11.53 The current i L for the network in Fig. 11.51.

43 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INSTANTANEOUS VALUES The development presented in Section 10.8 for capacitive networks can also be applied to R-L networks to determine instantaneous voltages, currents, and time. The instantaneous values of any voltage or current can be determined by simply inserting t into the equation and using a calculator or table to determine the magnitude of the exponential term.

44 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] AVERAGE INDUCED VOLTAGE: vL av For inductors, the average induced voltage is defined by:

45 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] AVERAGE INDUCED VOLTAGE: vL av FIG. 11.54 Current i L to be applied to a 4 mH coil in Example 11.8.

46 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] AVERAGE INDUCED VOLTAGE: vL av FIG. 11.55 Voltage across a 4 mH coil due to the current in Fig. 11.54.

47 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTORS IN SERIES AND IN PARALLEL FIG. 11.56 Inductors in series.

48 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTORS IN SERIES AND IN PARALLEL FIG. 11.57 Inductors in parallel.

49 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INDUCTORS IN SERIES AND IN PARALLEL FIG. 11.58 Example 11.9. FIG. 11.59 Terminal equivalent of the network in Fig. 11.58.

50 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS We found in Section 11.5 that, an ideal inductor can be replaced by a short-circuit equivalent once steady-state conditions. Recall that the term steady state implies that the voltage and current levels have reached their final resting value and will no longer change unless a change is made in the applied voltage or circuit configuration.

51 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS For all practical purposes, our assumption is that steady-state conditions have been established after five time constants of the storage or release phase have passed. FIG. 11.60 Substituting the short-circuit equivalent for the inductor for t > 5 t.

52 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS FIG. 11.61 Establishing the equivalent network for t > 5t.

53 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS FIG. 11.62 Example 11.10.

54 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS FIG. 11.63 Example 11.11.

55 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] STEADY-STATE CONDITIONS FIG. 11.64 Substituting the short-circuit equivalents for the inductors and the open-circuit equivalents for the capacitor for t > 5t.

56 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] ENERGY STORED BY AN INDUCTOR The ideal inductor, like the ideal capacitor, does not dissipate the electrical energy supplied to it. It stores the energy in the form of a magnetic field.

57 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] ENERGY STORED BY AN INDUCTOR FIG. 11.65 The power curve for an inductive element under transient conditions.

58 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] ENERGY STORED BY AN INDUCTOR FIG. 11.66 Example 11.12.

59 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS FIG.11.67 Dimmer control: (a) external appearance; (b) internal construction; (c) schematic.

60 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS FIG. 11.68 Basic operation of the dimmer in Fig. 11.67: (a) full voltage to the lamp; (b) approaching the cutoff point for the bulb; (c) reduced illumination of the lamp.

61 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS FIG. 11.69 Direct rheostat control of the brightness of a 60 W bulb.

62 Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim FIG. 11.74 Using Multisim to obtain the transient response for an inductive circuit.